An alternating electric field, of frequency v, is applied across the dees (radius =R) ofa cyclotron that is being used to accelerati protons (mass =m). The operating magnetic field (B) used in the cyclotron and the kinetic energy (K) of the proton beam, produced by it, are given by

Options

A. `B = (mv)/(e)` and `K = 2 m pi^(2) v^(2) R^(2)`
B. `B = (2pi m v)/(e)` and `K = m^(2) pi v R^(2)`
C. `B = (2pi m v)/(e)` and `K = 2m pi^(2) v^(2) R^(2)`
D. `B = (mv)/(e)` and `K = m^(2) pi v R^(2)`

Answer

C

Explanation

`T = (2pi m)/(eB) implies B = (2nm)/(eT) implies B = (2nm)/(e) ""v`
`R = (mv)/(eB) = (P)/(eB) implies P = eBR`
`implies P = e xx (2pi mv)/(e) = 2 pi mv R`
`k = (p^2)/(2m) = 2pi^(2) mv^(2) R^(2)`